<head>
<title>Mercator (1SP)</title>
</head>
<body>

<h1>Mercator (1SP)</h1>

<table border>

<td>Name
<td>Mercator
<tr>

<td>EPSG Code
<td>9804
<tr>

<td>GeoTIFF Code
<td>CT_Mercator (7)
<tr>

<td>OGC WKT
<td>Mercator_1SP
<tr>

<td>Supported By
<td>EPSG, GeoTIFF, PROJ.4, OGC WKT
<tr>

</table>

<h3>Projection Parameters</h3>

<table border>
<th>Name
<th>EPSG #
<th>GeoTIFF ID
<th>OGC WKT
<th>Units
<th>Notes

<tr>
<td>Latitude of natural origin
<td>1
<td>NatOriginLat
<td>latitude_of_origin
<td>Angular
<td>

<tr>
<td>Longitude of natural origin
<td>2
<td>NatOriginLong
<td>central_meridian
<td>Angular
<td>

<tr>
<td>Scale factor at natural origin
<td>5
<td>ScaleAtNatOrigin
<td>scale_factor
<td>Unitless
<td>

<tr>
<td>False Easting
<td>6
<td>FalseEasting
<td>false_easting
<td>Linear
<td>

<tr>
<td>False Northing
<td>7
<td>FalseNorthing
<td>false_northing
<td>Linear
<td>

</table>

<h3>Notes</h3>

I believe the formula (EPSG) notes are exactly the same for Mercator (1SP)
and <a href="mercator_2sp.html">Mercator (2SP)</a>.  There is only one projection in GeoTIFF for straight Mercator.<p>

<h3>PROJ.4 Organization</h3>

PROJ.4 does not support a latitude of natural origin other than the equator.   
<b>
<pre>
  +proj=merc  +lon_0=<i>Longitude of natural origin</i>
              +k_0=<i>Scale factor at natural origin</i> 
              +x_0=<i>False Easting</i>
              +y_0=<i>False Northing</i>
</pre>
</b>

A more common formulation for Mercator is to drop the +k_0, and instead to
provide a latitude of true scale using the +lat_ts parameter, which is the 
latitude at which the scale is 1.<p>

<h3>EPSG Notes</h3>

Mercator<p>
	
The Mercator projection is a special case of the Lambert Conic Conformal projection  
with the equator as the single standard parallel. All other parallels of latitude are straight 
lines and the meridians are also straight lines at right angles to the equator, equally spaced. 
It is little used for land mapping purposes but is in universal use for navigation charts and 
is the basis for the transverse and oblique forms of the Mercator. As well as being 
conformal, it has the particular property that straight lines drawn on it are lines of constant 
bearing. Thus navigators may derive their course  from the angle the straight course line 
makes with the meridians.
<p>

In the few cases in which the Mercator projection is used for terrestrial applications or 
land mapping, such as in Indonesia prior to the introduction of the Universal Transverse 
Mercator, a scale factor may be applied to the projection. This has the same effect as 
choosing two standard parallels on which the true scale is maintained at equal north and 
south latitudes either side of the equator.
<p>

<pre>
The formulas to derive projected Easting and Northing coordinates are:

For the two standard parallel case, k0 is first calculated from

k0 = cos*1/(1 - e2sin2*1)1/2 
 
where *1 is the absolute value of the first standard parallel (i.e. positive).  

Then, for both one and two standard parallel cases, 

	E = FE + ak0(* - *0)              
	N = FN + ak0 logn{tan(*/4 + */2)[(1 - esin*) / (1 + esin*)]e/2 } 
           		where symbols are as listed above and logarithms are natural.
	
The reverse formulas to derive latitude and longitude from E and N values are:
	
	* = * + (e2/2 + 5e4/24 + e6/12 + 13e8/360) sin(2*) 
		+ (7e4/48 + 29e6/240 + 811e8/11520) sin(4*)
		+ (7e6/120 +  81e8/1120) sin(6*)  + (4279e8/161280) sin(8*)

where 		* = */2 - 2 arctan t
		t = B (FN-N)/(ak0)
		B = base of the natural logarithm, 2.7182818...
		and  for the 2 SP Case, k0 is calculated as for the forward transformation 
above.
 
	* =  ((E - FE)/ak0)  + *0","For Projected Coordinate System Makassar / NEIEZ

Parameters:
Ellipsoid   Bessel 1841  a = 6377397.155 m   1/f = 299.15281
then e = 0.08169683

Latitude Natural Origin         00o00'00""N  = 0.0000000 rad
Longitude Natural Origin    110o00'00""E  = 1.91986218 rad
Scale factor ko                  0.997
False Eastings FE              3900000.00 m
False Northings FN              900000.00 m

Forward calculation for: 
Latitude            3o00'00.00""S   = -0.05235988 rad
Longitude     120o00'00.00""E   =  2.09439510 rad
gives
Easting  E   =      5009726.58 m
Northing N  =        569150.82 m

Reverse calculation for same easting and northing first gives :
t    = 1.0534121
chi = -0.0520110

Then Latitude     =   3o00'00.000""S
         Longitude  = 120o00'00.000""E
</pre>
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